Polarity–local times equivalence for Gaussian random fields

Let $X$ be a Gaussian random field satisfying Assumptions,, and. A point $z\in\mathbb{R}^d$ is polar for $X$ if $\mathbb{P}\{z\in X(I)\}=0$, where $I$ is the parameter set. Under these assumptions, the integral condition is

$$\int_0^{\delta_0} \frac{r^{N-1}}{\sigma^d(r)}\,dr=\infty.$$

Polarity–local times equivalence. Points are polar for $X$ if and only if

$$\int_0^{\delta_0} \frac{r^{N-1}}{\sigma^d(r)}\,dr=\infty.$$

For the Gaussian random fields considered here, this conjecture would extend the known equivalence between polarity of points and non-existence of local times for Lévy processes, additive Lévy processes, and fractional Brownian motion. The preceding theorem establishes the equivalence under the additional condition, but the general case remains open.